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MATHEMATICS: A. EMCH 
point sets which are in a (q, 2 p) — correspondence. Ci and C2 are 2 p-fold and 
q-fold curves of the surface. The surface has moreover p real and 2 pq — 2 p — 
q -]- 1 imaginary double generatrices. 
When q = 2s is even the generatrices cut Ci and C2 in two point sets which are 
in an also rational {s, p) — correspondence. Ci and Ci are respectively p- and 
s-fold curves of the surface. The surface has no real, hut ps — p — s -\- 1 imag- 
inary double generatrices. 
In the whole discussion the assumption is made, of course, that p and q 
are relatively prime. 
Theorem 3. When q is odd the order of each of the (q — l)/2 double curves is 
4 p ox 2 q according as q ^2 p. They are rational and each lies on a surface of 
revolution of order 4 generated by the rotation of an equilateral hyperbola about 
the z-axis. 
Theorem 4. When q = 2s is even and s odd, there are (s — l)/2 double curves 
of order 2p or q according as p ^ s, and one double curve of order p or s, accord- 
ing as p ^ s. When s = 2(t is even, there are a double curves of order 2 p or q, 
according as p ^ s. 
The intersections of the double curves with a plane through the z-axis may 
be arranged according to certain cyclic groups whose orders may be easily 
determined. One interesting fact is that the surfaces of the class in certain 
species, are applicable among themselves. The following theorems apper- 
tain to this fact: 
Theorem 5. Surfaces of the class are applicable to each other when their orders 
are 2 {p -\- q) and 2 (m p -\- n q), and the ratio of the radii of their Ci^s is m/n, 
with q odd, p and q, m and n, and m and q as relative primes. 
Theorem 6. Surfaces of the class of odd order are applicable to each other when 
their orders are p q and m p -\- n q, and the ratio of the radii of their Ci^s is 
m/n. Moreover q is even, and p and m are odd. 
As the surfaces of even and odd orders are respectively bifacial and uni- 
facial, we have 
Theorem 7. Bifacial and unifacial surfaces of the class are applicable to sur- 
faces of the same type only. 
The intersection of a torus with the surfaces of the class yields all so-called 
cycloharmonic curves. Also the 'bands of Moebius' may readily be cut out 
from the surfaces. 
Among the class here considered are cubic, quartic, and quintic scrolls in- 
vestigated by Cremona, Cayley, Schwarz and others. Models of these have 
been and are being constructed in the mathematical model shop of the 
University of Illinois. 
